Problem: Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{z^2 + 5z - 24}{z^2 - 4z + 3}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{z^2 + 5z - 24}{z^2 - 4z + 3} = \dfrac{(z + 8)(z - 3)}{(z - 1)(z - 3)} $ Notice that the term $(z - 3)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z - 3)$ gives: $a = \dfrac{z + 8}{z - 1}$ Since we divided by $(z - 3)$, $z \neq 3$. $a = \dfrac{z + 8}{z - 1}; \space z \neq 3$